2009-08-24 On cylindrically symmetric codes Divakar Viswanath talking to Predrag, about Nicholas Grisouard and Oliver Bühler code for "Bose-Einstein condensate" on a 2-dimensional disk:

• Bessel ⇒ Hankel transform ⇒ inverse Hankel transform is a complicated code. The main expense is in quadratures. One also needs to precompute and tabulate the inverses. I could show Grisouard how to speed it up some.
• I follow Nick Trefethen, Spectral Methods in MATLAB, Chapter 11 and use Chebyshev polynomials. So does Rich Kerswell. Fourier and Chebyshev quadratures are essentially for free. Chebyshevs could be numerically unstable if too many points are used radially (200 or more), but so far no problem - in pipe simulations we use 10-20 radial point

2009-09-10 On cylindrically symmetric codes Dwight Barkley talking to Predrag, about numerics in axially symmetric flows:

• In my long pipe calculations with David Moxley I use SEMTEX spectral element DNS code. I do not like Chebyshevs in pipe simulations because they are very wasteful around the center of the pipe, where not many points are needed.

2009-11-10 Predrag, on quotienting M/SO(2) from 2D sections of 3D flow Do you guys want to skype later today or tomorrow? We'll discus here how to slice pipe experimental data, and I am stuck thinking in the state space, have no good idea how to slice SO(2) given only the full measured time dependent 3D velocity field in a lab-stationary 2-dim section of the downstream moving 3D pipe flow…

2009-12-01 Marc to Ashley The problem with the slice was that when the Newton failed the routine getz did not manage to locate the shift. To convert the S3, I did the following:

1. Compute eigenvalues.
2. Add +-0.01*v1 (here v1 is the leading eigenvalue)
3. Double the domain (Here program compiled with alpha=1.25 and doubled i_K to keep resolution).
4. Bring to Mp=1. (Here I compiled with i_Mp=3 and multiplied i_M by 3 to keep resolution).
5. Problem is that after this I get relaminarisation straight away for the two cases. Wondering if doing something completely wrong.
6. I'll try to continue S3 to k=1.25 and see what happens.

Also, I observed that the Arnoldi never returns eigenvalues with non-zero imaginary part. I would expect at least some very small number, indicating that it's zero to the computed precision. Any ideas on this?

2009-12-01 Predrag Need to make some strategic decisions.

• Use TW database of R.R. Kerswell, O.R. Tutty Recurrence of Travelling Waves in Transitional Pipe Flow
• pick canonical Re, pipe length. Current plan is to use Kerswell and Tutty values
  

Re = 2400, L_z = 5 D

• to start with, pick TW's from their tables closest to the turbulent flow
• pick TW's within all symmetry subspaces
• pick preferentially least unstable TW's.
• pick some pairs TW≠σTW
• include some rotational TW's (doubly travelling)
• slice azymuthal O(2), to avoid having to find the symmetry lines of solutions fixed under {e,σ}.

2009-12-01 Ashley to Rich We've got my code to project a state onto a hyperplane which contains a reference TW and the origin. We're lacking TWs at the moment though and were discussing where would be best to start, to narrow it down to a reasonable small list of TWs to request from you. We've looked at Kerswell & Tutty 2007. From table 2 it looked like well visited solutions are

• 3a_2.5
• 3b_2.5
• 2a_1.25
• 2b_1.25

two of which are upper and the other two lower branch. I hope it's not too difficult to find them. Any solution which rotates (but is not helical) would also be very handy to check projections which involve θ-shifts.

2009-12-01 Rich

• K & Tutty 2007 is a good place to start
• 5D long pipe is the way to go
• Re = 2000-2800 should cover it well
• TWs I can easily give you all those we dealt with there (37?) as I have them bundled up for Tutty.
• One important point - Pringle, C.C.T. & Kerswell, R.R. ``Asymmetric, helical and mirror-symmetric travelling waves in pipe flow'' Phys. Rev. Lett. 99, 074502, 2007 and Pringle, C.C.T., Duguet, Y. & Kerswell, R.R. ``Highly symmetric travelling waves in pipe flow'' Phil. Trans. Roy. Soc. A 367, 457-472, 2009 describe what I believe are more important TWs (occur earlier in Re and are more symmetric!). I suggest reading the Phil Trans paper and having a think until Friday..

2009-12-07 Rich, Ashley, Marc, Predrag & Björn

• important are nicely symmetric “Nigel's” - Rich will prepare N3
• 'Mothers' might also be important
• (will continue - now culture calls)

2010-05-11 Ashley to Marc I've had a chance to come back to the projection work. Please let me know if I've made a mistake in the following.

Let x be any TW already projected into a plane using a reference state. (I've just used a randomly chosen turb state.) Then we take

 e1 = xp + Z xp
 e2 = xp - Z xp

where Z is the reflection operator (about th=0), and xp is a TW used for projection within the plane. Rich's N,M states have shift & reflect about th=0, and mirror sym about some other plane. I don't want to rotate xp back s.t. it satisfies xp = Z xp, as then e2=0. (I've checked that this can be done with the TWs though.) The projections of the TWs within the slice are

 x.hat{e1}
 x.hat{e2} .

Basically I'm finding that e1 and e2 are an awful choice for projection, perhaps I'm using the wrong definitions?:

• x.e1 is dominated by the component of x independent of the z-shift.
• x.e2 is identically zero for all TWs with different alpha or m0 to that of xp.

Any ideas?

2010-05-28 Cycling in a tight enclosure

Working in m=2 + shift&reflect + shift&rotate here… There are four N2 states at this wavelength. Using the leading complex eigenvalue for the lower middle state (N2_M1) the corresponding eigenvectors have been used for projection with in the slice. N2_M1 has also been used for the reference to define the slice, it fills all coeffs within the subspace.

The lower state is on the boundary between the attraction to the laminar state (bottom right). Not bad for a day's cycling. Ash.

2010-05-30 Predrag to Ashley: You are on the roll

looks very promising! And no, channelflow.org is not set up to send emails out, so send alerts out by your email, in active periods one clicks on [Recent changes] button, upper right to see whether there are any new edits.

2010-05-31 Rich Any chance of some more detail on the webpage as to how the cyan, purple and black trajectories are initiated (I presume there are 3 separate orbits shown here?). Also what are the 4 blue blobs? (N3s with different alphas?) - pipe length? What is N2_M1?

2010-05-31 Predrag to Rich: Ashley and Marc can answer the above questions. Ashley also found a tiny relative periodic orbit close (and presumably bifurcated off) the attracting equilibrium in the upper left corner. My proposal is to work at the same chubby fat pipe length L=1.8 D and Reynolds Re = 2200 (?) as Fernando, not be restricted to the shift-rotate invariant subspace (that is what is in the figure above), and also go to m's higher than m=2. One will lose the tow travelling waves in the middle and the tiny relative periodic orbit, but figuring out how the unstable manifolds bracket the (transiently?) turbulent region of the state space will be easier, and that is the main thing to work out now; going to longer pipes will come later.

2010-06-04 Ash to Rich, Predrag: The blue dots from top-left to bottom-right are the N2_125_U, M2, M1, L states. The pink dot is the laminar state. The M2 and L states have real leading eigenvalues and the black and cyan lines are started with perturbations +- the leading eigenvector. The M1 state has complex leading eigenvalue, so spirals out. I think the spiral appears squashed as I expect the eigenvectors have components in the direction of the reference tangent vector. The RPO Predrag mentions is nice in that it has a much shorter period than the others that have bifurcated of a TW (approx 5D/U), as far as I know. Note that it doesn't encircle the TW in the projection, but that could also be due to the slicing?

As we've already got states for L=2.5D, Re=2400, I think we should try opening up a symmetry first - it might not be necessary to change parameters yet… Marc also suggests trying ICs in the other parts of the phase space, but keeping the current symmetries, in order to see if there's turbulence elsewhere.

2010-06-04 Predrag OK, stay here, open up the symmetry…

2010-07-08 Ash Had a go at removing the shift-and-rotate symmetry. The remaining symmetries are m=2 + shift&reflect. It'd be cool if you've some S2 states at this α=1.25. The L state still has one real unstable direction, but the M1,M2 and U states have new leading complex eigenvalues. Shooting in these new directions…

On the lhs it shows that the 'turbulence' explores much more of the space than before, but not so close to the N2 TWs. I wonder where S2 TWs are?.. On the rhs I've projected again relative to M1, but using the new (symmetry-breaking) leading eigenvector. I can then see spiraling out nicely, but all the TWs are now on the origin. [2010-09-16 Ash: added S2 states to left plot in green]

2010-07-09 Humble(dt) “[…] all the TWs are now on the origin” Probably both the TWs and the leading eigenvectors still have too much symmetry. How does the world look from your older, no-symmetry turbulent snapshot vantage point? Roger (from Adriatic).

2010-07-10 Marc We have started an exploration of the turbulent behavior of the 2.5D pipe. This consists of computing lifetimes (at Re=2400, probably others as well) for different symmetry restrictions: we start with full space, then m=2, Shift & Reflect and Rotate & Reflect are progressively added. The idea is to see how the transients get shorter and even if 'turbulence' is at all possible with the highly constrained cases. Additionally, we'll run correlations of the DNS lifetime data with exact states to see if this may aid the search for connections among states. Hopefully this will provide an overview/characterization of the system we are trying to visualize and understand!

2010-07-10 Humble(dt) 2 Mothers & Nigels Roger!

2010-09-16 Ash See updated entry on 2010-07-08. “[…] all the TWs are now on the origin” (right-hand plot). The TWs have no component in the direction of the symmetry breaking eigenvector, seen to within the numerical precision of the root finding of for the shifts at least. Relative to the turbulent snapshot we've had difficulties choosing axis for projection within the slice. For the moment I can remove the last (S&Refl) symmetry and check we still have transients, otherwise it might be worthwhile starting to look for close recurrences. Also I can check where the unstable directions of the S2 states go. Each has one unstable direction, real for the lower state, complex for the upper. It would be nice to have a hetroclinic connection between the (S&Rot) N2 states and the S2 states, but the N2 states have quite a few symmetry-breaking unstable directions to search through…

2010-09-16 Humble(dt) 2 Ashley It's probably a sin to replace an earlier plot in the blog by a more obscure one. In any case, can you replace 1857m1proj.png by something bigger (of order of 800-1000 pixels across) so one can look at it more closely? You can separately set its size in the blog, so that one gets the big version by clicking on it.

2010-09-16 Ash 2 Humble(dt) I'm a habitual sinner - try clicking on the image now.

2010-11-23 Ash Been looking for recurrences in the turbulent cloud (entry 2010-07-08). Its somewhat hampered by lots of jumps in the z-shift, S_z. In Fourier space the tangent to x is generated by t_a=Tx, where T=diag{im}. I've experimented with altering the definition just for the reference tangent t'=diag{i.m^n}x'. More negative n helps but isn't great (see plot). I've had a look at the equations for dS_z/dt but haven't thought of a reference tangent which that could avoid the jumps (i.e. avoids g t_a . t'=0). Have there been any developments for the KS case? Marc, any suggestions from your recent simulations for where this search may be fruitful?

2010-11-23 Humble(dt) 2 Ashley I think Stefan Froehlich and I have now a clean presentation of how slicing works and precise statement how the reduced flow jumps through these singularities, but this is currently on siminos CNS subversion repository. If you guys have intellectual surplus to follow the latest slice & dice breakthroughs, I will have to make you jump a few firewall hoops. Have done it with Marc, but I suspect he has given up on reading these internal blogs. OK, telegraphic version: slicing is the only game in town, and will work, but you will have to construct multiple charts (slices centered on qualitatively different neighborhoods in the state space).

2010-11-24 Ash 2 Humble(dt) I think I can see why there are likely to be jumps; by jumping through these, are you suggesting there's a particular slice it should jump to? I'll have a go at tracking a few slices at once…

2010-11-30 Humbledt 2 Ashes Try this - take two slice fixing points ('reference states'), each for a qualitatively different but frequently visited neighborhood (a Mother and a Nigel?). The two slices are hyperplanes, so their intersection is also a hyperplane of a lower dimension. Swithc from slice to slice whenever you cross that hyperplane. We have not implemented this yet for Kuramoto-Sivashinsky, so I might be underestimating the difficulty of implementation, but hopefully it is a simple linear programming test, and you do not have to be precise about when you switch the slices - if we are lucky, all singularities are someplace further out on each slice that the boundary between them.

2011-01-12 Humbledt 2 plumbers Dear Masters of Pipes, I've labored incredibly long to write a short intro to slicing for Ashley and whoever has the courage to try it, and here is a readable draft, "Reduction of continuous symmetries of chaotic flows by the method of slices" by Stefan Froehlich and Predrag Cvitanović. I tried to make it a few pages, but failed - would be grateful to anyone who tells me what more to cut/move to appendices, etc.

Anyway, it's my firm belief that we can do it, so let's get to it.

2011-01-17 Ash Following the entry on 2010-11-30, I've tried using all our TWs (for m=2, L=2.5D) as reference states and keep track the z-shift for each, S_i(t), i=1..7 (1 Laminar; 2 LB; 3 M1; 4 M2; 5 UB; 6 S2a; 7 S2b). For the plot below I start with index iref=3, and when S_iref crosses another S_i, I switch iref if we're closer to state i. There are certainly fewer jumps in dS_iref/dt.

It picks up an excursion towards the LB state quite nicely, and for turbulence we find it hanging around S2a (see Kerswell & Tutty 07), but I'm still struggling to find orbits. How should I pick the shift of the templates? Having read Predrag's draft, maybe I should repeat adding, say, 100 randomly chosen turbulent states as templates?

2011-01-17 Humbledt 2 plumbers No - as few slices as possible. And there should be no jumps in dS_z/dt, none at all. We just need to make sure that the ridges between the templates are sufficiently close to each the templates, so that the inflection hyperplanes are excluded. Once templates are picked, the rest is geometry of hyperplanes (NOTHING to do with dynamics, only with the group theory) so checking whether the inflection hyperplane is on the far side of the tile edge (ridge between two slices) is a linear computation, to be undertaken independently of dynamics. I hope…

For my answers to Ashley's comments on the Slice & dice paper, click here (the problem is that if one is writing LaTeX, it's much easier to use siminos subversion blog, rather than copying stuff to here - dokuwiki is clunky when it comes to LaTeX.

Regarding the latest figures - I usually think that time series in dynamics are not too insightful, would be nice to also see the reduced state space projections of the flow on one slice (for now), with segments of trajectory where dS_z/dt is large color-coded. Might see more clearly where the inflection hyperplane lies. I am bit worried about the shift velocities that you do see - I would hope there is a closer template in each case. so the singularity is preempted by switching to its neighborhood.

2011-02-16 Ash Using the set of TWs as templates, I've been able to get quite long tracks without jumps. This has meant that I could do some recurrence checking… Here's another RPO found using the slicing! [pink], T=45 R/(2U)=11.25 D/U; the crosses are equally spaced in time. (S&Rot symmetry has been removed.) The candidate for the long-period orbit I've found difficult to get to converge.. I'm trying multiple shooting at the mo'.

2011-02-16 Humbledt you are an angel, but I cannot see what the left figure is. Mea culpa, no doubt. Join us for Marc's Webinar, or be square.

2011-02-17 Ash Sorry I missed the meeting, give me a bit longer warning next time. I've registered but not found it online - is it recorded anywhere? Re yesterday's entry: The figure on the LHS shows the relative distance within the slice between states separated by time dt. The signature around t=3100..3300 suggested that the trajectory shadowed an orbit four or five times. When plotted, the candidate trajectory looked like it might just be spiralling away from the S2a state (green spot on RH figure), but it proved worthwhile trying with Newton. The plotted orbit was located within half a dozen iterations.

2011-03-31 Ash Second RPO added to figure on 2011-02-16, the longer one (blue). T=37.93 D/U; marks on trajectories are 2 D/U apart. The orbit wanders roughly equidistant from M1 (lower blue circle) and S2a (green spot), not just around S2a as this particular projection suggests.

2011-04-14 Humbledt Wow! this one will require some thinking. In the spirit of Froehlich & Cvitanovic paper, we will have to add a Poincare section hyperplane through each template point, start looking into local segments of the unstable manifolds to start working out symbolic dynamics for RPOs and making sense of it all. Already the (blue) RPO is too convoluted to interpret without making a stab at symbolic dynamics of longer orbits.

2011-04-08 Ash And another (orange). Very similar to the first RPO (pink), but having compared energies and friction factors, they are not just different projections of the same orbit.

2011-05-29 Humbledt Got this link form Mason Porter, it might be helpful to non-Brits in this collaboration.

2011-05-29 Humbledt Starting this date, the blog has been moved to CNS svn repository `pipes'.